Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1506 conjugacy classes of subgroups; only the 255 normal subgroups are shown.
This group has 1506 conjugacy classes of subgroups — too many to draw. Only its 255 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C2×#1113 |
| commutator | C22×C33 | C22×C4 |
| Fitting, prime radical | C24×C33 | C22 |
| socle, cosocle | C23×C33 | C23 |
| hypercenter | C24 | #1113 |
| hypocenter | C33 | #381 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C23 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #11722082 | C2 | semidirect |
| 8. C22×C3×C3⋊Dic3 | C4 | semidirect |
| 38. C22×C3⋊Dic3 | Dic3 | semidirect |
| 184. C33 | #381 | semidirect |
| 297. C32 | #4943 | semidirect |
| 298. C32 | #5211 | semidirect |
| 341. C3 | #10591696 | semidirect |
| 342. C3 | #10591697 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×C3⋊Dic3 | 2 |
| C2×C3×#3757 | 2 |
| #11722082 | 2 |
| #10591697 | 3 |
| #10591696 | 3 |